Self-affine Sets Research Articles

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Abstract Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.

The dimensions of inhomogeneous self-affine sets

Funding: SAB thanks the Carnegie Trust for financially supporting this work. JMF was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1).

DOUBLING PROPERTIES OF SELF-AFFINE MEASURES SUPPORTED ON GATZOURAS–LALLEY FRACTALS

In this paper, we study the doubling properties of self-affine measures supported on a class of self-affine sets named Gatzouras–Lalley sets. First, we show that every self-affine measure is doubling if the Gatzouras–Lalley sets satisfy “strong separation” in both horizontal and vertical directions. Then we provide a sufficient condition for self-affine measures supported on Gatzouras–Lalley sets to be doubling if the sets do not satisfy the “separation condition” in the horizontal direction. Finally, we investigate measures without “vertical separation”, and give sufficient conditions for them to be doubling.

Dimensions of equilibrium measures on a class of planar self-affine sets

We study equilibrium measures (Käenmäki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier–Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of a coordinate projection of the measure. In particular, we do this by showing that the Käenmäki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.

The Hausdorff dimension of integral self-affine sets through fractal perturbations

The estimation or the computation of the Hausdorff dimension of self-affine fractals is of considerable interest. An almost-sure formula for that has been given by K.J. Falconer. However, the precise Hausdorff dimension formulas have been given only in special cases. This problem is even unsolved for a general integral self-affine set F, which is generated by an n×n integer expanding matrix T (not necessarily a similitude) and a finite set A⊂Rn of integer vectors so that F=T−1(F+A). In this paper, we focus on the pivotal case F⊂R2 and show that the Hausdorff dimension of F is the limit of a monotonic sequence of McMullen-type dimensions by introducing a process, which we call the fractal perturbation (or deflection) method. In fact, the perturbation method is developed to deal with the pathological case where the characteristic polynomial of T is irreducible over Z. We also consider certain examples of exceptional self-affine fractals for which the Hausdorff dimension is less than the upper bound given by Falconer's formula. These examples show that our approach leads to highly non-trivial computation.

An infinitely generated self-similar set with positive Lebesgue measure and empty interior

In [Problems on self-similar sets and self-affine sets: An update, Birkhäuser, Basel, 2000] Peres and Solomyak ask the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Csörnyei et al. in 2006. The authors of that paper gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They did not however provide an explicit example. Motivated by a desire to construct an explicit example, we provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.

An explicit formula for the pressure of box-like affine iterated function systems

In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the “box-like” self-affine fractals investigated by D.-J. Feng, Y. Wang, and J. M.‘Fraser. Combining previous results of V. Yu. Protasov, A. Käenmäki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J. M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.

On thin carpets for doubling measures

We study thin sets for doubling or isotropic doubling measures in R d \mathbb {R}^{d} . In our results we prove that the self-affine sets satisfying the open set condition with holes are thin for isotropic doubling measures and among them Barański carpets are thin for doubling measures.

Hausdorff dimension of planar self-affine sets and measures

Let $$X={\bigcup }{\varphi }_{i}X$$ be a strongly separated self-affine set in $${\mathbb {R}}^2$$ (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the $$\varphi _{i}$$ , we prove that $$\dim X$$ is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.

Characterization of a class of planar self-affine tile digit sets

Characterization of a class of planar self-affine tile digit sets

On the Dimension and Measure of Inhomogeneous Attractors

A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with notable results by Olsen, Snigireva, Fraser and Käenmäki on inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous self-affine sets. This paper is devoted to filling a significant gap in the dimension theory of inhomogeneous attractors, by studying those formed from arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension of the condensation set and a quantity related to pressure, which we term upper Lipschitz dimension, forms a natural and general upper bound on the dimension. Additionally, we begin a new line of enquiry; the methods developed are used to classify the Hausdorff measure of inhomogeneous attractors. Our results have applications for affine systems with affinity dimension less than or equal to one and systems satisfying bounded distortion, such as conformal systems in dimensions greater than one.

Digit frequencies and self-affine sets with non-empty interior

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\unicode[STIX]{x1D6FD}\in (1,1.787\ldots )$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a simply normal $\unicode[STIX]{x1D6FD}$-expansion. We also prove that if $\unicode[STIX]{x1D6FD}\in (1,(1+\sqrt{5})/2)$ then every $x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$ has a $\unicode[STIX]{x1D6FD}$-expansion for which the digit frequency does not exist, and a $\unicode[STIX]{x1D6FD}$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1$, then every non-trivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and gives rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Barany, Hochman- Solomyak and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

We study operators defined on a Hilbert space defined by a self-affine Delone set $$\Lambda $$ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain $$\limsup $$ law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or $${\mathbb {R}}^d$$ -invariant distributions of a dynamical system defined by $$\Lambda $$ . We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on $$\ell ^2(\Lambda )$$ and show that the errors of convergence in Shubin’s formula are given by these traces.

Dimension of generic self-affine sets with holes

Let (Sigma , sigma ) be a dynamical system, and let Usubset Sigma . Consider the survivor set ΣU=x∈Σ∣σn(x)∉Ufor alln\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Sigma _U=\\left\\{ x\\in \\Sigma \\mid \\sigma ^n(x)\\notin U\\,\\text {for all}\\,n\\right\\} \\end{aligned}$$\\end{document}of points that never enter the subset U. We study the size of this set in the case when Sigma is the symbolic space associated to a self-affine set Lambda , calculating the dimension of the projection of Sigma _U as a subset of Lambda and finding an asymptotic formula for the dimension in terms of the Käenmäki measure of the hole as the hole shrinks to a point. Our results hold when the set U is a cylinder set in two cases: when the matrices defining Lambda are diagonal; and when they are such that the pressure is differentiable at its zero point, and the Käenmäki measure is a strong-Gibbs measure.

Dimension of self-affine sets for fixed translation vectors

An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by B\'ar\'any and K\"aenm\"aki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self-affine sets and measures in any Euclidean space.

Self-affine sets in analytic curves and algebraic surfaces

We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.

Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

A fundamental problem in the dimension theory of self-affine sets is the construction of high-dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high-dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. Whilst the existence of these equilibrium states has been well-known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three-dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self-affine set in $\mathbb{R}^3$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension.

On the dimension of triangular self-affine sets

As a continuation of a recent work [Bárányet al, On the dimension of self-affine sets and measures with overlaps.Proc. Amer. Math. Soc.144(2016) 4427–4440] of the same authors, in this note we study the dimension theory of diagonally homogeneous triangular planar self-affine iterated function systems.

CONNECTEDNESS OF SELF-AFFINE SETS WITH PRODUCT DIGIT SETS

Let [Formula: see text] be an expanding lower triangular matrix and [Formula: see text]. Let [Formula: see text] be the associated self-affine set. In the paper, we generalize some connectedness results on self-affine tiles to self-affine sets and provide a necessary and sufficient condition for [Formula: see text] to be connected.